Path planning can be implemented as a two-step process:                1. find a collision-free, static path through the world; and        2. find a speed profile that is safe and takes vehicle limitations into account.        
Path planning methodologies, for example using Mixed Integer Linear Programming (MILP), are known for determining globally optimal paths for vehicles between a starting position and a desired terminal position. For example, International Patent Application no. PCT/GB2012/050041 (Publication No. WO2012/098375) describes a trajectory planning system for vehicles, in which the trajectory is determined using a linear approximation of the dynamics of the vehicle, which linear approximation is constrained by requirements that acceleration of the vehicle during the trajectory is less than a threshold value for the acceleration, and the threshold value is dependent on an infinity norm of the velocity of a vehicle. These constraints aim to take into account a notion of vehicle heading, which ensures that the determined trajectory does not feature motion that cannot be achieved by conventional wheeled or tracked vehicles.
The use of linear approximation in path planning methodologies is advantageous in that it simplifies and enables rapid path planning, whilst minimising processing power required and also ensuring that the methodology can be relatively easily integrated into larger systems. However, this and other path planning methodologies, facilitate profiling over a path for a ground vehicle, but tend not to be ideal for path planning in respect of airborne vehicles because they do not take into account additional constraints in relation to height and factors affecting climb performance. These factors require the determination of speed and height profiles, which makes the problem non-linear and, therefore, much more complex.
In more detail, and referring to FIGS. 6(a) to (c) of the drawings, for a given route of 2D waypoints, arrival time, speed and height values must be found in order to perform path planning. However, as shown in the Figures, each waypoint has a position with bounds on its height and bounds on the arrival time such that acceleration, velocity and climb rate constraints are satisfied; and optimisation of this problem is strongly non-linear with its key variables being coupled by non-linear equations.
The present invention seeks to address at least some of these issues and provide a method and system for rapid path profiling and, therefore, simplified path planning in respect of airborne vehicles.